Quantum Groupoids

Xu, Ping

doi:10.1007/s002200000334

Cited by 150 publications

(248 citation statements)

References 33 publications

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“…The Lie algebroid generalization of the sheaf of L-poly-differential operators is denoted by D L poly (X) [27], [3]. It is the tensor algebra over O of the universal enveloping algebra of L (see §3.3 below).…”

Section: Gerstenhaber Algebras and Precalculi By Definition A Gerstementioning

confidence: 99%

“…The algebra (better: in the terminology of [27], [3] "the Hopf algebroid") U R (L) may be thought of as an algebra of L-differential operators on R. In the case L = Der k (R) and R smooth over k then U R (L) coincides with the algebra of differential operators on R. 5 Note that there is, at first sight, a more natural right R-module structure on UR(L) given by the formula Dr = Di(r). This alternative right module structure will not be used in this paper.…”

Section: L-connectionsmentioning

confidence: 99%

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Căldăraru's conjecture and Tsygan's formality

2012

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In this paper we complete the proof of Cȃldȃraru's conjecture on the compatibility between the module structures on differential forms over polyvector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects.Our methods use formal geometry to globalize the local formality quasiisomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact -recently proved by the first two authors -that Shoikhet's quasi-isomorphism is compatible with cap products after twisting with a Maurer-Cartan element.

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Section: Gerstenhaber Algebras and Precalculi By Definition A Gerstementioning

confidence: 99%

Section: L-connectionsmentioning

confidence: 99%

Căldăraru's conjecture and Tsygan's formality

2012

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“…In addition to Lu's mapping above, define bialgebroid arrows A e → S, a ⊗ b → s L (a)t L (b) and the Xu anchor mapping S → End A given by x → ε(?s L (x)). P. Xu's anchor map [23] corresponds to the action of S on A via source and counit [14, 3.7], for which A becomes the unit module in the tensor category of S-modules. …”

Section: Discussionmentioning

confidence: 99%

Pseudo-Galois Extensions and Hopf Algebroids

Kadison

Modules and Comodules

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Abstract. A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is nontrivially isomorphic to a Connes-Moscovici Hopf algebroid, which raises interesting comparative questions.

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“…The R-bialgebroid structure of R#U L projects to VL (see [38]), and by [22, §4.2.1] one also has that VL carries a left Hopf algebroid structure, the translation map on generators a ∈ R, X ∈ L of VL being given by…”

Section: Lie Algebroids and Associated Left Hopf Algebroidsmentioning

confidence: 99%

Morita Base Change in Hopf-Cyclic (Co)Homology

Kaoutit

Kowalzig

2012

Lett Math Phys

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ABSTRACT. In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.

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Quantum Groupoids

Cited by 150 publications

References 33 publications

Căldăraru's conjecture and Tsygan's formality

Căldăraru's conjecture and Tsygan's formality

Pseudo-Galois Extensions and Hopf Algebroids

Morita Base Change in Hopf-Cyclic (Co)Homology

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